To get the \(\mathcal {O}\left(\frac{1}{c}\right)\)  correction one needs to construct \(\mathcal {P}_{h,\bar{h}}\)  accordingly. For light external operators (\(\Delta _{i}\) ) this has been carried out with considerable mathematical rigour in [1]} for \(\langle \chi (\infty )\chi (1)\mathcal {O}(x)\mathcal {O}(0)\rangle \)  configuration. This can be connected to \(\langle \chi (\infty )\chi (0)\mathcal {O}(1)\mathcal {O}(1-x)\rangle \)  via the transformation \(z\rightarrow \frac{x-1}{z-1}\) . The expression for vacuum block was derived in [2]} and can also be independently verified from the result for general \(h\)  in [1]}
\(\langle \chi \vert \mathcal {O}(t_{euc}/\tau )\mathcal {O}(0)\vert \chi \rangle &=&(1-x)^{h_{\mathcal {O}}}\langle \chi (0)\mathcal {O}(1)\mathcal {O}(1-x)\chi (\infty )\rangle \nonumber \\&=&(1-x)^{h_{\mathcal {O}}}x^{-2h_{\mathcal {O}}}\left[ 1+\frac{2h_{\chi }h_{\mathcal {O}}x^{2}}{c} {_{2}}F_{1}(2,2,4,x)\right]\nonumber \\&=&\left(\sinh \frac{\pi t}{\beta }\right)^{-2h_{\mathcal {O}}}\left[1-\frac{24 h_{\chi }h_{\mathcal {O}}}{c}\left(1-\frac{\pi t}{\beta }\coth \frac{\pi t}{\beta }\right)\right]\nonumber \\ \nonumber \) 
